Semigroup Forum

, Volume 89, Issue 1, pp 77–104 | Cite as

On semigroups of endomorphisms of a chain with restricted range

  • Vítor H. Fernandes
  • Preeyanuch Honyam
  • Teresa M. Quinteiro
  • Boorapa Singha
RESEARCH ARTICLE

Abstract

Let X be a finite or infinite chain and let \({\mathcal{O}}(X)\) be the monoid of all endomorphisms of X. In this paper, we describe the largest regular subsemigroup of \({\mathcal{O}}(X)\) and Green’s relations on \({\mathcal{O}}(X)\). In fact, more generally, if Y is a nonempty subset of X and \({\mathcal{O}}(X,Y)\) is the subsemigroup of \({\mathcal{O}}(X)\) of all elements with range contained in Y, we characterize the largest regular subsemigroup of \({\mathcal{O}}(X,Y)\) and Green’s relations on \({\mathcal{O}}(X,Y)\). Moreover, for finite chains, we determine when two semigroups of the type \({\mathcal {O}}(X,Y)\) are isomorphic and calculate their ranks.

Keywords

Transformations Order-preserving Restricted range Rank 

References

  1. 1.
    Adams, M.E., Gould, M.: Posets whose monoids of order-preserving maps are regular. Order 6, 195–201 (1989) MATHMathSciNetCrossRefGoogle Scholar
  2. 2.
    Aĭzenštat, A.Ya.: The defining relations of the endomorphism semigroup of a finite linearly ordered set. Sib. Mat. Zh. 3, 161–169 (1962) (Russian) Google Scholar
  3. 3.
    Aĭzenštat, A.Ya.: Homomorphisms of semigroups of endomorphisms of ordered sets. Uč. Zap.—Leningr. Pedagog. Inst. 238, 38–48 (1962) (Russian) Google Scholar
  4. 4.
    Aĭzenštat, A.Ya.: Regular semigroups of endomorphisms of ordered sets. Uč. Zap.—Leningr. Gos. Pedagog. Inst. 387, 3–11 (1968) (Russian) Google Scholar
  5. 5.
    Almeida, J., Volkov, M.V.: The gap between partial and full. Int. J. Algebra Comput. 8, 399–430 (1998) MATHMathSciNetCrossRefGoogle Scholar
  6. 6.
    Fernandes, V.H.: Semigroups of order-preserving mappings on a finite chain: a new class of divisors. Semigroup Forum 54, 230–236 (1997) MATHMathSciNetCrossRefGoogle Scholar
  7. 7.
    Fernandes, V.H.: Semigroups of order-preserving mappings on a finite chain: another class of divisors. Russ. Math. (Izv. VUZ) 3(478), 51–59 (2002) (Russian) MathSciNetGoogle Scholar
  8. 8.
    Fernandes, V.H., Sanwong, J.: On the rank of semigroups of transformations on a finite set with restricted range. Algebra Colloq. (to appear) Google Scholar
  9. 9.
    Fernandes, V.H., Volkov, M.V.: On divisors of semigroups of order-preserving mappings of a finite chain. Semigroup Forum 81, 551–554 (2010) MATHMathSciNetCrossRefGoogle Scholar
  10. 10.
    Fernandes, V.H., Jesus, M.M., Maltcev, V., Mitchell, J.D.: Endomorphisms of the semigroup of order-preserving mappings. Semigroup Forum 81, 277–285 (2010) MATHMathSciNetCrossRefGoogle Scholar
  11. 11.
    Gomes, G.M.S., Howie, J.M.: On the ranks of certain semigroups of order-preserving transformations. Semigroup Forum 45(3), 272–282 (1992) MATHMathSciNetCrossRefGoogle Scholar
  12. 12.
    Hall, M. Jr.: Combinatorial Theory. Wiley, New York (1967) MATHGoogle Scholar
  13. 13.
    Higgins, P.M.: Divisors of semigroups of order-preserving mappings on a finite chain. Int. J. Algebra Comput. 5, 725–742 (1995) MATHMathSciNetCrossRefGoogle Scholar
  14. 14.
    Howie, J.M.: Products of idempotents in certain semigroups of transformations. Proc. Edinb. Math. Soc. 17, 223–236 (1971) MATHMathSciNetCrossRefGoogle Scholar
  15. 15.
    Howie, J.M.: Fundamentals of Semigroup Theory. Oxford University Press, Oxford (1995) MATHGoogle Scholar
  16. 16.
    Jitjankarn, P.: Isomorphism Theorems for Semigroups of Order-preserving Full Transformations (2012). arXiv:1202.2977v1 [math.RA]
  17. 17.
    Kemprasit, Y., Changphas, T.: Regular order-preserving transformation semigroups. Bull. Aust. Math. Soc. 62(3), 511–524 (2000) MATHMathSciNetCrossRefGoogle Scholar
  18. 18.
    Kim, V.I., Kozhukhov, I.B.: Regularity conditions for semigroups of isotone transformations of countable chains. Fundam. Prikl. Mat. 12(8), 97–104 (2006) (Russian). Translation in J. Math. Sci. (N.Y.) 152(2), 203–208 (2008) Google Scholar
  19. 19.
    Mendes-Gonçalves, S., Sullivan, R.P.: The ideal structure of semigroups of transformations with restricted range. Bull. Aust. Math. Soc. 83, 289–300 (2011) MATHMathSciNetGoogle Scholar
  20. 20.
    Mora, W., Kemprasit, Y.: Regular elements of some order-preserving transformation semigroups. Int. J. Algebra 4(13–16), 631–641 (2010) MATHMathSciNetGoogle Scholar
  21. 21.
    Nenthein, S., Youngkhong, P., Kemprasit, Y.: Regular elements of some transformation semigroups. Pure Math. Appl. 16(3), 307–314 (2005) MATHMathSciNetGoogle Scholar
  22. 22.
    Repnitskiĭ, V.B., Vernitskii, A.: Semigroups of order preserving mappings. Commun. Algebra 28(8), 3635–3641 (2000) MATHCrossRefGoogle Scholar
  23. 23.
    Repnitskiĭ, V.B., Volkov, M.V.: The finite basis problem for the pseudovariety \(\mathcal{O}\). Proc. R. Soc. Edinb., Sect. A, Math. 128, 661–669 (1998) MATHCrossRefGoogle Scholar
  24. 24.
    Sanwong, J., Sommanee, W.: Regularity and Green’s relations on a semigroup of transformations with restricted range. Int. J. Math. Math. Sci. (2008). Art. ID 794013, 11 pp. Google Scholar
  25. 25.
    Sanwong, J., Singha, B., Sullivan, R.P.: Maximal and minimal congruences on some semigroups. Acta Math. Sin. Engl. Ser. 25(3), 455–466 (2009) MATHMathSciNetCrossRefGoogle Scholar
  26. 26.
    Sullivan, R.P.: Semigroups of linear transformations with restricted range. Bull. Aust. Math. Soc. 77, 441–453 (2008) MATHMathSciNetCrossRefGoogle Scholar
  27. 27.
    Symons, J.S.V.: Some results concerning a transformation semigroup. J. Aust. Math. Soc. A 19, 413–425 (1975) MATHMathSciNetCrossRefGoogle Scholar
  28. 28.
    Vernitskii, A., Volkov, M.V.: A proof and generalisation of Higgins’ division theorem for semigroups of order-preserving mappings. Izv. Math. Izv. Vuzov Matematika 1, 38–44 (1995) (Russian) MathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • Vítor H. Fernandes
    • 1
    • 2
  • Preeyanuch Honyam
    • 3
  • Teresa M. Quinteiro
    • 2
    • 4
  • Boorapa Singha
    • 5
  1. 1.Departamento de Matemática, Faculdade de Ciências e TecnologiaUniversidade Nova de LisboaCaparicaPortugal
  2. 2.Centro de Álgebra da Universidade de LisboaLisboaPortugal
  3. 3.Department of Mathematics, Faculty of ScienceChiang Mai UniversityChiang MaiThailand
  4. 4.Instituto Superior de Engenharia de LisboaLisboaPortugal
  5. 5.School of Mathematics and Statistics, Faculty of Science and TechnologyChiang Mai Rajabhat UniversityChiang MaiThailand

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